The invention relates generally to inertial navigation systems and more specifically to sensors used in such systems to measure the values of environmental variables and the compensation of such sensors for variations in other environmental variables.
Strapdown inertial navigation systems are frequently used in missiles and aircraft. Physically isolated and stabilized apparatus, such as a gimballed platform that is physically angularly-stabilized relative to the local vertical direction, require precise and mechanically complex angle positioning apparatus, and are being systematically replaced by systems of the strapdown type.
A state-of-the-art strapdown inertial navigation system has three rotation sensors or gyros and three accelerometers rigidly attached to a supporting vehicle. The rotation sensors are each positioned and oriented to sense angular displacement about one of three defined orthogonal axes attached to the vehicle body and known as the body coordinate system. The accelerometers are each positioned and oriented in a fixed direction relative to the vehicle, to sense velocity changes (incremental velocities) along different ones of the three defined orthogonal axes. In a strapdown system, the accelerometer axes are not angularly stabilized.
Because the accelerometers are constantly changing direction relative to gravity, navigation velocities cannot be computed by directly integrating the accelerometer signals. Instead, a stable computational frame or analytic navigation coordinate system is continually generated. The output signals from the rotation sensors are used by an attitude integration apparatus to calculate the directions of local vertical, together with two other axes orthogonal to the local vertical direction.
Sensed angle changes and accelerations (incremental velocities) are continually rotated through the calculated angles from the vehicle body axes to the calculated navigation axes. Angle signals from the rotation sensors are used to update the computer-stored angular position and incremental velocity data for both the angle sensors and accelerometers relative to the navigation coordinate system.
The rotation sensors and accelerometers have fixed relative directions in the body coordinate system. An angular transformation matrix of direction cosines is computed in an attitude integration apparatus. The accelerometer signals, which are incremental changes in velocity, in the strapdown body coordinate system are converted in a coordinate transformation computer from that system into corresponding signals in the stabilized navigation coordinate system.
After transformation into the navigation coordinate system, the incremental velocity signals are integrated or summed to form updated velocity signals. The rotation sensor and accelerometer signals are sampled, and the sampled signals are delivered to a computer which is programmed to accept the signals and to calculate both velocities along the three axes in the stabilized navigation coordinate system and attitude angles relative to this system.
The estimate x.sub.1e produced by a sensor of an environmental variable x.sub.1 may be affected by the values of other environmental variables x.sub.2, x.sub.3, . . . , x.sub.K such as temperature, pressure, and humidity and possibly first and even higher-order time derivatives of such variables. To achieve the ultimate measurement accuracy, the output x.sub.1e of such a sensor is compensated for variations in these other environmental variables by means of a compensation model .delta.x.sub.1 (x.sub.1e, x.sub.2e, x.sub.3e, . . . , x.sub.Ke) where x.sub.1e, x.sub.2e, x.sub.3e, . . . , x.sub.Ke are the outputs from sensors that result respectively from the sensor inputs x.sub.1, x.sub.2, x.sub.3, . . . , x.sub.K. The compensated output x.sub.1c is given by EQU x.sub.1c =x.sub.1e -.delta.x.sub.1 (1)
Note that the compensation model .delta.x.sub.1 may be a function of x.sub.1. The compensation model can be expressed as ##EQU1##
The quantity x.sub.ie denotes a particular set of the variables of interest x.sub.1e, x.sub.2e, x.sub.3e, . . . , x.sub.Ke. The set may be different for each value of i. The function .function..sub.i (x.sub.ie) is a function of the one or more variables in the set x.sub.ie associated with the index i. The symbols a.sub.i denote quantities that are constant, at least over the short term.
We can rewrite equation (1) as EQU x.sub.1c -x.sub.1a =x.sub.1e -x.sub.1a -.delta.x.sub.1 (3)
where x.sub.1a is the actual value of x.sub.1 supplied by an external source. We wish to choose the coefficients a.sub.i so that x.sub.1c is an accurate estimate of x.sub.1. We do this by finding the values which minimize a statistical measure of the magnitude of ##EQU2##
for a variety of values for x.sub.1, x.sub.2, x.sub.3, . . . , x.sub.K. A suitable statistical measure might be the sum of the squares of ##EQU3##
for the variety of values for x.sub.1, x.sub.2, x.sub.3, . . . , x.sub.K.
The sensor outputs may be noisy and it may be desirable for the purpose of minimizing the computational requirements to apply a noise-reducing operator G to the sensor outputs prior to determining the values of a.sub.i. The operator G transforms the values of a sensor output associated with times extending from present time minus a predetermined time interval T to present time into a single present-time value. An example of an operator G operating on a function x(t) is provided by a lowpass filter which is described mathematically by the equation ##EQU4##
where the function h(t) is the impulse response of the filter and is essentially zero for times greater than T. For some cases, the operator G may be nonlinear: G(x.sub.1 +x.sub.2) may not be equal to the sum of Gx.sub.1 and Gx.sub.2.
Applying G to equation (3), EQU G(x.sub.1c -x.sub.1a)=G(x.sub.1e -x.sub.1a -.delta.x.sub.1) (5)
A relatively simple approach to determining a compensation model is to assume G to be reasonably linear. We also assume Gx.sub.ke to be equal to x.sub.ke at times T after a change in the values of one or more of the sensor inputs x.sub.1, x.sub.2, . . . , x.sub.k, . . . , x.sub.K in the absence of noise. Then at such times T, ##EQU5##
where the notation Gx.sub.ie means that each x.sub.ke in the functional representation is replaced by Gx.sub.ke.
We determine the values of the coefficients a.sub.i by finding the values which minimize a statistical measure of the magnitude of ##EQU6##
using the values obtained at times T after changes in the values of x.sub.i.
This process for determining the a's may be difficult, inconvenient, and inadequate for several reasons. First of all, the process is time-consuming since x.sub.1, x.sub.2, x.sub.3, . . . , x.sub.K must be held constant for time intervals T. Second, the calibration must sometimes include variables that are time derivatives of other variables. One cannot hold a variable constant if the time derivative of the variable is required to take on values other than zero.
A calibration process is needed that does not require the variables affecting the output of a sensor to be held constant for time periods T during the execution of the calibration process.